Is there an equation I can use to calculate the temperature (as a function of time) of an object which is gaining or losing heat by convection? Or equivalently, the rate of energy transfer from the object to the surrounding fluid (or vice-versa)? It can involve constants representing properties of the object and the surrounding fluid.

@Marek: hm, interesting. I glanced over those before, but taking a second look I notice that at the latter link, it says there are formulas to calculate the heat transfer coefficient $h$ for certain systems. I'd be very interested to see information about hose formulas posted here, if you (or anyone else) is inclined to do so.
–
David Z♦Jan 1 '11 at 7:37

I am afraid that I don't know anything about these topics besides the general overview. But I too am very eager to learn more, so I hope someone else will elaborate.
–
MarekJan 1 '11 at 14:04

Heh, well if we could get by without squiggles we'd be on english.SE. (though: it missed "wordos"? :-P) Anyway I was hoping for a simple formula that wouldn't require the flow rate or temperatures but I'm not too surprised to hear that there isn't one.
–
David Z♦Dec 29 '10 at 8:19

Haha; that makes a lot more sense now! Funnily enough the "or" didn't occur to me else I would have corrected it myself.
–
NoldorinDec 29 '10 at 15:47

here you can find some formulas for calculating Nusselt, Prandtl, Reynolds, Rayleigh and Grasshoff numbers. Those are important for evaluating conditions in different systems. Numbers will tell you which state of convection is around your geometry (natural, forced, laminar, turbulent, external, internal). For each case there exist some correlation.

More info about Nusselt number and others you can find eg. on Wikipaedia.

If you were interested in the temperature distribution within a solid in contact with a fluid (e.g. gas or liquid), you would solve the heat equation, e.g.:
$$ \frac{\partial T}{\partial t} = k\nabla^2T$$. The loss of heat via convection to the surrounding fluid would show up in the choice of boundary conditions and specifically you would use the Robin Condition (or mixed condition) that is a balance of conduction in the solid and convection in the fluid at the interface and can be derived from applying Newton's Law of Cooling at the interface. For example at the boundary $L$:
$$ k\frac{\partial T(t,L)}{\partial n}+h T(t,L)=g(t)$$ where $k$ is the thermal conductivity in the solid and $h$ is the convective heat transfer coefficient for the fluid.

If you were interested in the temperature variation within a fluid with convective heat transfer, you want to solve a more general heat equation that contains a convective term that then couples its solution with a solution for the motion of the fluid (e.g. Navier Stokes),
$$ \frac{\partial T}{\partial t} + \tilde{u} \cdot \nabla T= k\nabla^2T$$ where $\tilde{u}$ is the velocity vector (thus making this a vector equation).

There are some formulas (for example, for cylinders, such as pipes) in the following book: Bosworth R.C.L. Heat transfer phenomena. The flow of heat in physical systems. – Associated general publications, PTY. LTD. – Sydney. – 1952, 211 p. Both heat conduction (in the so called stagnant film near the pipe) and convection are important for pipe cooling in the air. It is interesting that the relevant coefficients depend on the orientation of the pipes (vertical/horizontal).